Attempt 2
If the density of solid iron is
7.87
g/cm
3
, what is the packing efficiency of
Fe
if it adopts a body‑centered cubic unit cell? The molar mass of
Fe
is
55.847
g/mol
.
783
To calculate the packing efficiency of iron (Fe) in a body-centered cubic unit cell, we need to determine the volume of the atoms in the unit cell and the volume occupied by the atoms.
First, let's find the volume of one Fe atom in the unit cell:
1. Determine the length of the unit cell edge (a).
In a body-centered cubic unit cell, the body diagonal (d) is equal to 4 times the atomic radius (r), which can be expressed as:
d = 4r
2. Rearrange the equation to isolate r:
r = d / 4
3. Substitute the values:
d = a√3 (Using the Pythagorean theorem since it's a body-centered cubic unit cell)
a√3 = 4r
a = 4r / √3
Now, we have the value of 'a' in terms of the atomic radius 'r'.
4. Find the volume of the Fe atom:
The volume (V) of a sphere with radius 'r' can be calculated using the formula:
V = (4/3)πr^3
5. Substitute the value of 'r' into the formula to calculate the volume of the Fe atom.
Once we have the volume of one Fe atom, we need to find the volume occupied by the atoms in the unit cell:
6. Determine the number of atoms per unit cell:
In a body-centered cubic unit cell, there is one whole atom at each corner and one atom at the center. Therefore, the number of atoms (n) is:
n = 1 + 1/8
= 1.125
7. Multiply the number of atoms by the volume of each atom to get the total unit cell volume occupied by the atoms:
Volume occupied by atoms (V_total) = n * V
Finally, we can calculate the packing efficiency:
8. Calculate the packing efficiency:
Packing Efficiency = (Total volume occupied by atoms / Total volume of the unit cell) * 100
To summarize, follow these steps:
1. Calculate 'a' using the equation a = 4r / √3, with 'r' as the atomic radius.
2. Calculate the volume of one Fe atom using V = (4/3)πr^3.
3. Calculate the number of atoms per unit cell (n) as 1 + 1/8.
4. Calculate the total volume occupied by the atoms (V_total) by multiplying 'n' with the volume of one Fe atom.
5. Calculate the packing efficiency using the formula (V_total / total volume of the unit cell) * 100.
Make sure to use the given density of solid iron (7.87 g/cm^3) and molar mass of Fe (55.847 g/mol) to determine the atomic radius (r).